CONTENTS xi
12.9. Examples of Passage from the Semisimple Case 290
12.10. Reductive Commutative Spaces 293
Chapter 13. Structure of Commutative Nilmanifolds 299
13.1. The "2-step Nilpotent" Theorem 299
13.1 A. Solvable and Nilpotent Radicals 299
13.1B. Group Theory Proof 300
13.1C. Digression: Riemannian Geometry Proof 301
13.2. The Case Where N is a Heisenberg Group 303
13.3. The Chevalley-Vinberg Decomposition 309
13.3A. Digression: Chevalley Decompositions 309
13.3B. Weakly Commutative Spaces 314
13.3C. Weakly Commutative Nilmanifolds 317
13.3D. Vinberg's Decomposition 318
13.4. Irreducible Commutative Nilmanifolds 319
13.4A. The Irreducible Case Classification 320
13.4B. The Irreducible Case Structure 321
13.4C. Decomposition into Irreducible Factors 326
13.4D. A Restricted Classification 327
Chapter 14. Analysis on Commutative Nilmanifolds 329
14.1. Kirillov Theory 329
14.2. Moore-Wolf Theory 330
14.3. The Case where N is a (very) Generalized Heisenberg Group 335
14.4. Specialization to Commutative Nilmanifolds 338
14.5. Spherical Functions 341
14.5A. General Setting for Semidirect Products N x K 342
14.5B. The Commutative Nilmanifold Case 342
Chapter 15. Classification of Commutative Spaces 345
15.1. The Classification Criterion 345
15.2. Trees and Forests 350
15.2A. Trees and Triples 350
15.2B. The Mixed Case 351
15.2C. The Nilmanifold Case 353
15.3. Centers 354
15.4. Weakly Symmetric Spaces 357
Bibliography 367
Subject Index 373
Symbol Index 383
Table Index 387
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